EXPECTATION FORMULAS
7
27. CHI-SQUARE DISTRIBUTION
Begin with standard normal Z and let Z1 , Z2 , Z3 , ., Zd be an independent random sample
of size d. Dene Wd and 2 by
d
2
d
d
X
2
= fW 2 , Wd =
2
Zk , mW 2 (t) = (1
d /2
2t)
k=1
If X is normal and IRS is
THE EXPECTATION PRIMER
EXPECTATION COVARIANCE AND PROBABILITY V3.2
7
that one way to arrive at a guessing procedure for expectation is to try to guess the long run
average, in the case of random variables. In either case, we therefore see that the rst fou
SETS & FUNCTIONS
3
3.6. SET INTERSECTION:. If A and B are sets, then their Intersection, denoted
A \ B, is the set
A \ B = cfw_x 2 A : x 2 B = cfw_x : x 2 A & x 2 B = cfw_x 2 B : x 2 A,
and thus consists of their common membership. The symbol \ is calle
SETS & FUNCTIONS
7
5.7. AXIOM OF CHOICE:. If C is a non-empty collection of non-empty sets, then
Y
C 6= ;.
5.8. DISTRIBUTIVE LAWS:. For any sets A, B, C,
A \ ( B [ C ) = (A \ B ) [ ( A \ C ) ,
A [ ( B \ C ) = (A [ B ) \ ( A [ C ) ,
A ( B \ C ) = (A B ) \
SETS & FUNCTIONS
9
5.20. MUTUALLY INVERSE FUNCTIONS:. We say that f and g are Mutually
Inverse Functions provided that g f is the identity on the domain of f and f g is
the identity on the domain of g. Thus g f and f g are identity functions. In more
deta
4
M. J. DUPRE
This will mean that in case of a continuous unknown, we can approximate to whatever
degree of accuracy we might require with unknowns which are discrete (notice again that
Ln (X ),Rn (X ), and Un (X ) only assume at most a countable innity o
THE EXPECTATION PRIMER
EXPECTATION COVARIANCE PROBABILITY V3.2
MAURICE J. DUPRE
1. INTRODUCTION
These notes are designed to complement an elementary course in probability and/or statistics. The aim is to develop all the mathematical tools required using o
SETS & FUNCTIONS
MAURICE J. DUPRE
1. SETS
1.1. SET:. an undened term. A set S can also be called a Collection.
1.2. SET MEMBERSHIP:. The statement that x is a member of set S is denoted x 2 S.
We also say x is in S to mean x 2 S. We write x 2 S to mean th
EXPECTATION FORMULAS
11
Suppose now we have X1 , X2 , X3 , ., XnX and Y1 , Y2 , Y 3, ., YnY are independent random
samples for each random variable, so XnX or simply X is the sample mean random variable
for the population A, and likewise YnY or simply Y i
EXPECTATION FORMULAS
Maurice J. Dupr
e
Department of Mathematics
New Orleans, LA 70118
email: [email protected]
November 2010
In the formulas given below, U, W, X, Y are any unknowns, and A, B, C, D, K, are any statements, unless otherwise specied. E (X |
EXPECTATION FORMULAS
3
11. DEFINITION OF COVARIANCE
X ] [ Y
Cov (X, Y ) = E ([X
Y ])
12. DEFINITION OF VARIANCE
V ar(X ) = Cov (X, X )
13. DEFINITION OF STANDARD DEVIATION
p
SD(X ) = X = V ar(X )
14. DEFINITION OF CORRELLATION
Cov (X, Y )
X Y
= (X, Y ) =
EXPECTATION FORMULAS
23. DIRAC DELTA FUNCTION
Not really a function but it is denoted
with compact support
h(0) =
with the property that for any smooth function h
Z
1
h(x) (x)dx
1
c ( x)
h( c ) =
5
= (x
Z
1
c)
h(x) c (x)dx
1
M( c )(t) = ect
If A1 , A2 , A
EXPECTATION FORMULAS
9
30. SAMPLING TO ESTIMATE THE MEAN
If X is a random variable with known standard deviation X but with unknown mean, not a
common circumstance, then in case Xn is normal, a condence interval for the mean can easily
be given for any Le
MODELO DE VALUACION Y RIESGO
INSTRUCCIONES
Con la informacin de la Compaa EL CAFETAL, que se proporciona realice lo siguiente:
a) El Flujo de Efectivo del proyecto para 10 aos
b) El Estado de Resultado proyectado para 10 aos
c) El Balance General para los